Geometric Sequence Calculator
Calculate the nth term, sum of first n terms, and infinite sum of any geometric sequence with step-by-step solutions and interactive visualization.
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About Geometric Sequence Calculator
Welcome to our Geometric Sequence Calculator, a powerful mathematical tool that calculates the nth term, sum of the first n terms, and infinite sum of any geometric sequence. Whether you are studying mathematics, preparing for exams, or solving real-world problems involving exponential growth or decay, this calculator provides accurate results with detailed step-by-step solutions and interactive visualizations.
What Is a Geometric Sequence?
A geometric sequence (also called a geometric progression) is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio (r). This multiplicative pattern distinguishes geometric sequences from arithmetic sequences, where terms differ by a constant addition.
For example, the sequence 3, 6, 12, 24, 48, ... is geometric because each term is twice the previous term (r = 2). The sequence 100, 50, 25, 12.5, ... is also geometric with r = 0.5, showing how terms can decrease.
Key Components of a Geometric Sequence
- First term (a₁): The starting value of the sequence
- Common ratio (r): The constant multiplier between consecutive terms
- nth term (aₙ): Any specific term at position n in the sequence
- Sum (Sₙ): The total of the first n terms
Geometric Sequence Formulas
The nth Term Formula
To find any term in a geometric sequence, use the formula:
Where a₁ is the first term, r is the common ratio, and n is the position of the term. The exponent is (n-1) because we multiply by r zero times to get the first term, once to get the second term, and so on.
Sum of First n Terms
The sum of the first n terms depends on whether the common ratio equals 1:
When r = 1, all terms are equal, so Sₙ = n × a₁.
Infinite Sum (Convergent Series)
When |r| < 1, the terms approach zero and the infinite sum converges:
If |r| ≥ 1, the series diverges and has no finite sum.
How to Use This Calculator
- Enter the first term (a₁): Input the starting value of your geometric sequence. This can be positive, negative, or a decimal.
- Enter the common ratio (r): Input the value by which each term is multiplied. The ratio can be positive, negative, or fractional.
- Enter n: Specify which term position you want to find, and how many terms to sum.
- Select precision: Choose the number of decimal places for your results (10-100).
- Click Calculate: View the nth term, sum, sequence visualization, and step-by-step solution.
Understanding Sequence Behavior
Growth vs Decay
- Growth (r > 1): Terms increase without bound. Example: 2, 6, 18, 54, ... (r = 3)
- Decay (0 < r < 1): Terms decrease toward zero. Example: 100, 50, 25, ... (r = 0.5)
- Oscillating (-1 < r < 0): Terms alternate signs and decrease in magnitude. Example: 8, -4, 2, -1, ... (r = -0.5)
- Oscillating Growth (r < -1): Terms alternate signs and increase in magnitude. Example: 2, -6, 18, -54, ... (r = -3)
- Constant (r = 1): All terms equal the first term. Example: 5, 5, 5, 5, ...
- Alternating Constant (r = -1): Terms alternate between +a₁ and -a₁. Example: 7, -7, 7, -7, ...
Real-World Applications
Finance & Investment
Compound interest calculations, where money grows by a fixed percentage each period, follow geometric sequence patterns. An investment growing at 8% annually multiplies by 1.08 each year.
Biology & Population
Bacterial growth, where cells divide at regular intervals, follows geometric progression. If bacteria double every hour, the population follows a sequence with r = 2.
Physics & Engineering
Radioactive decay, sound intensity reduction, and signal attenuation follow geometric decay patterns where each interval reduces the quantity by a constant factor.
Computer Science
Algorithm complexity analysis often involves geometric series. Binary search halves the problem size each step, and recursive algorithms frequently exhibit geometric patterns.
Frequently Asked Questions
What is a geometric sequence?
A geometric sequence (or geometric progression) is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio (r). For example, 2, 6, 18, 54, ... is a geometric sequence with first term a₁=2 and common ratio r=3.
What is the formula for the nth term of a geometric sequence?
The nth term of a geometric sequence is given by the formula: aₙ = a₁ × r^(n-1), where a₁ is the first term, r is the common ratio, and n is the position of the term you want to find. For example, if a₁=3 and r=2, the 5th term is a₅ = 3 × 2^4 = 48.
How do you find the sum of a geometric sequence?
The sum of the first n terms of a geometric sequence is Sₙ = a₁(1-rⁿ)/(1-r) when r≠1, or Sₙ = n×a₁ when r=1. For an infinite geometric series where |r|<1, the sum converges to S∞ = a₁/(1-r).
When does a geometric series converge?
A geometric series converges (has a finite sum to infinity) when the absolute value of the common ratio is less than 1 (|r| < 1). This means terms get progressively smaller and approach zero. If |r| ≥ 1, the series diverges and has no finite sum.
What is the difference between geometric and arithmetic sequences?
In an arithmetic sequence, each term differs from the previous by a constant amount (common difference). In a geometric sequence, each term is a constant multiple (common ratio) of the previous term. Arithmetic: 2, 5, 8, 11 (add 3). Geometric: 2, 6, 18, 54 (multiply by 3).
Additional Resources
- Geometric Sequences - Mathematics LibreTexts
- Geometric Progression - Wikipedia
- Geometric Series - Wikipedia
Reference this content, page, or tool as:
"Geometric Sequence Calculator" at https://MiniWebtool.com/geometric-sequence-calculator/ from MiniWebtool, https://MiniWebtool.com/
by miniwebtool team. Updated: Jan 20, 2026
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